Sure it is. When you take a French course they have you learn the name "Etats-Unis" (or "Angleterre" or "Australie" or wherever you're from) as a French word along with all the others.
Yes. The name “Beijing” is for people writing in pinyin, which is not a transliteration designed for English speakers. English speakers should use Yale romanization or the traditional postal romanization for well known places.
It’s not. Beijing is pretty accurate as most English speakers say it (except for the tones), although I often hear people do really weird things with the j. The b and j sounds in Mandarin are not the same as in English, but they are closer than p or k to my ear.